Cyclicity in the Drury-Arveson space and other weighted Besov spaces

Abstract

Let H \mathcal {H} be a space of analytic functions on the unit ball B d {\mathbb {B}_d} in C d \mathbb {C}^d with multiplier algebra M u l t ( H ) \mathrm {Mult}(\mathcal {H}) . A function f ∈ H f\in \mathcal {H} is called cyclic if the set [ f ] [f] , the closure of { φ f : φ ∈ M u l t ( H ) } \{\varphi f: \varphi \in \mathrm {Mult}(\mathcal {H})\} , equals H \mathcal {H} . For multipliers we also consider a weakened form of the cyclicity concept. Namely for n ∈ N 0 n\in \mathbb {N}_0 we consider the classes C n ( H ) = { φ ∈ M u l t ( H ) : φ ≠ 0 , [ φ n ] = [ φ n + 1 ] } . \begin{equation*} \mathcal {C}_n(\mathcal {H})=\{\varphi \in \mathrm {Mult}(\mathcal {H}):\varphi \ne 0, [\varphi ^n]=[\varphi ^{n+1}]\}. \end{equation*} Many of our results hold for N N :th order radially weighted Besov spaces on B d {\mathbb {B}_d} , H = B ω N \mathcal {H}= B^N_\omega , but we describe our results only for the Drury-Arveson space H d 2 H^2_d here. Letting C s t a b l e [ z ] \mathbb {C}_{stable}[z] denote the stable polynomials for B d {\mathbb {B}_d} , i.e. the d d -variable complex polynomials without zeros in B d {\mathbb {B}_d} , we show that a m p ; if d is odd, then C s t a b l e [ z ] ⊆ C d − 1 2 ( H d 2 ) , and a m p ; if d is even, then C s t a b l e [ z ] ⊆ C d 2 − 1 ( H d 2 ) . \begin{align*} &\text { if } d \text { is odd, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d-1}{2}}(H^2_d), \text { and }\\ &\text { if } d \text { is even, then } \mathbb {C}_{stable}[z]\subseteq \mathcal {C}_{\frac {d}{2}-1}(H^2_d). \end{align*} For d = 2 d=2 and d = 4 d=4 these inclusions are the best possible, but in general we can only show that if 0 ≤ n ≤ d 4 − 1 0\le n\le \frac {d}{4}-1 , then C s t a b l e [ z ] ⊈ C n ( H d 2 ) \mathbb {C}_{stable}[z]\nsubseteq \mathcal {C}_n(H^2_d) . For functions other than polynomials we show that if f , g ∈ H d 2 f,g\in H^2_d such that f / g ∈ H ∞ f/g\in H^\infty and f f is cyclic, then g g is cyclic. We use this to prove that if f , g f,g extend to be analytic in a neighborhood of B d ¯ \overline {{\mathbb {B}_d}} , have no zeros in B d {\mathbb {B}_d} , and the same zero sets on the boundary, then f f is cyclic in ∈ H d 2 \in H^2_d if and only if g g is. Furthermore, if the boundary zero set of f ∈ H d 2 ∩ C ( B d ¯ ) f\in H^2_d\cap C(\overline {{\mathbb {B}_d}}) embeds a cube of real dimension ≥ 3 \ge 3 , then f f is not cyclic in the Drury-Arveson space.